Calculus Examples

$lim_{x \to - 100000} \frac{\sqrt{4 x^{2} - 1}}{x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 100000$ .

$\frac{lim_{x \to - 100000} \sqrt{4 x^{2} - 1}}{lim_{x \to - 100000} x}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to - 100000} 4 x^{2} - 1}}{lim_{x \to - 100000} x}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 100000$ .

$\frac{\sqrt{lim_{x \to - 100000} 4 x^{2} - lim_{x \to - 100000} 1}}{lim_{x \to - 100000} x}$

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{4 lim_{x \to - 100000} x^{2} - lim_{x \to - 100000} 1}}{lim_{x \to - 100000} x}$

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{4 {(lim_{x \to - 100000} x)}^{2} - lim_{x \to - 100000} 1}}{lim_{x \to - 100000} x}$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $- 100000$ .

$\frac{\sqrt{4 {(lim_{x \to - 100000} x)}^{2} - 1 \cdot 1}}{lim_{x \to - 100000} x}$

Step 7

Evaluate the limits by plugging in $- 100000$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $- 100000$ for $x$ .

$\frac{\sqrt{4 {(- 100000)}^{2} - 1 \cdot 1}}{lim_{x \to - 100000} x}$

Step 7.2

Evaluate the limit of $x$ by plugging in $- 100000$ for $x$ .

$\frac{\sqrt{4 {(- 100000)}^{2} - 1 \cdot 1}}{- 100000}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 100000$ to the power of $2$ .

$\frac{\sqrt{4 \cdot 10000000000 - 1 \cdot 1}}{- 100000}$

Step 8.1.2

Multiply $4$ by $10000000000$ .

$\frac{\sqrt{40000000000 - 1 \cdot 1}}{- 100000}$

Step 8.1.3

Multiply $- 1$ by $1$ .

$\frac{\sqrt{40000000000 - 1}}{- 100000}$

Step 8.1.4

Subtract $1$ from $40000000000$ .

$\frac{\sqrt{39999999999}}{- 100000}$

Step 8.2

Move the negative in front of the fraction.

$- \frac{\sqrt{39999999999}}{100000}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{39999999999}}{100000}$

Decimal Form:

$- 1.99999999 \dots$